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Decline of permeate flux for ultrafiltration along membrane tubes
DESALINATION Desalination 145 (2002) 153-l 57
ELSEVIER
www.elsevier.com/locate/desal
Decline of permeate flux for ultrafiltration along membrane tubes Ho-Ming Yeh Department of Chemical Engineering, Tamkang University Tamsui, Taipei, Taiwan 2.51, ROC Tel. +886 (02) 2621-5656, ext. 2601; Fax +886 (02) 262-0987; email: [email protected] Received 8 February 2002; accepted 22 February 2002
Abstract The decline of permeate flux for ultrafiltration along membrane tubes was predicted from mass and momentum balances, based on the resistance-in-series model coupled with the considerations of the declinations of flow rate and transmembrane pressure along the membrane tubes. It is concluded that the declination of permeate flux occurs mainly due to the decline of transmembrane pressure and the increase of the thickness of concentration polarization layer along the membrane tube, and thus it is rather sensitive for lower transmembrane pressure and larger solution concentration. Kcyworak
Ultrafiltration; Permeate flux declination; Membrane tube
1. Introduction The ultrafiltration (UF) process was initially developed for the treatment of waste waters and sewage to remove particulate and macromolecular materials. Nowadays, it is applied in a wide variety of fields, from the chemical industry (including electrocoat paint recovery, latex processing, textile size recovery and recovery of lubricating oil) to medical applications (such as kidney dialysis operation), and even to biotechnology (including concentration of milk, egg white, juice, pectin and sugar, and recovery of Presented at the International July 7-12, 2002. 001 l-9164/02/$-
Congress on Membranes
protein from cheese whey, animal blood, gelatin and glue) [ 1,2].
2. Theory A number of mathematical models for permeation analysis of membrane UF are available in the literature that attempt to describe the mechanism oftransport through membranes. The permeate flux of UF of micromolecular solutions is usually analyzed by following models: the gel polarization model [3-91, the osmotic pressure and Membrane
Processes
See front matter 0 2002 Elsevier Science B.V. All rights reserved
PII: SO0 I1 -9 I64(02)0040
I-O
(ICOA4,),Toulouse, France,
154
H.-M. Yeh / Desalination I45 (2002) 153-l 5 7
model [10-l 81,and the resistance-in-series model [19-261. 2.1. Resistance-in-series model In the resistance-in-series model, permeate flux v,(z) may be expressed as
tube, it can be assumed that the momentum balance for laminar flow inside the membrane tube is simply given by the Hagen-Poiseuille equation [27]
_dAp -=dz
vm(z) =
&p(z) R, +Rf+ 4 Ap(z)
(1)
In Eq. (l), R, denotes the intrinsic resistance, and $Ap(z> and Rf are, respectively, the resistance due to the concentration polarization/gel layer with 4 as a proportional constant [22,23] and those due to other fouling phenomena such as solute adsorption, while Ap(x) is the transmembrane pressure defined as
8w 7c
rf
where lo is the viscosity of solution. 2.4. Permeate flux The distribution of permeate flux v,(z) was derived by solving Eqs. (l), (3) and (4) with the following inlet conditions: q=ql
and
Ap=Apiatz=O
(5), (6)
The result is [26] Ap(z) =p(z)-P,
(2)
In the above equation, p(z) is the pressure distribution of the tube side along the axial direction, z, andp, is the permeate pressure of the shell side which may be assumed to be a constant. The resistances R, and R/as well as the proportional constant 4 in Eq. (1) are usually determined experimentally.
Ap
v=
(7)
1 +pAP
in which AP is determined from
2.2. Mass balance Consider a membrane tube of inside radius rm and tube length L. Let q(z) be the volumetric flow rate of solution flowing in the tube side and v,(z) be the permeate flux by UF. Then, a mass balance over a slice dz of the tube gives
dq= -2n; Y,V, dz
2.3. Momentum balance Since the permeation rate of solvent through the membrane wall is small compared with the volumetric flow rate of solution flowing in the
and V and AP are defined as v=
6pm+Rf)v=‘, ‘P A,p=
APi where
Gi
(9), (IO)
H. -M. Yeh / Desalination 145 (2002) 153-1.57
3. Discussion
and conclusions
For calculation of the permeate flux of water from aqueous dextran T500 through a tubular ceramic membrane, let us use the correlation equation of $ and (R, + R/> derived by Dong
P81. 4 (s/m) = 1.37 x 10’ uiWo.02 Cz~‘s3s
(15)
membrance pressure (Api>,solution concentration (C,), volumetric flow rate (q,), and tube radius (r,) as parameters, were calculated from Eq. (7), and some of the results are shown in Figs. 14, respectively. It is seen in Fig. 1 that v,,, declines more rapidly for lower Api due to the serious decline of the effective driving force (transmembrane pressure) in the downstream part of the tube. As shown in Fig. 2, v,,,declines rather 1.5;
(R,,,+ Rf)(Pa-s/m)
155
.
= 1.055~ 10”
.
“__. ._. ___-.-
L
+.__
_l...-_..”_...... -..-_l_ll_~
_
.
c,
=Ecl WI% ..__ _._
.‘
(16) + 1.74 x 10’0 uq~W5Cf.244 where the feed velocity U,is related to the volumetric flow rate of feed solution q, as u, = qilnr,jf. The correlation predictions for the distribution of permeate flux were calculated from Eq. (7) with the use of the empirical equation for the viscosity of dextran T500 aqueous solution [29]. ~1(kg/m’s) = 0.894x 10e3 exp(0.408Ci)
f
“+,
0.5
%rmw%; \
\
/
i /
(17)
‘.. _?(
.‘\\i
=I
‘*
0.0’
The decline of permeate flux v,(z) along the flow channel of tubular membranes with trans-
tcxltPR q,ts2%*w*tn3.+
1,=4Dcm r, =0,3mt 0.2
0.0
0.4
2:
0.6
0.8
l.C
Fig. 2. Effect of Ci on the decline of v,,,. 2.40,~ .__..._...._... .._._._._..__._ _.
2 "E 12 3 ", Y E
i 0.0’
i
0.0
.
0.2
i...
i.
0.4
0.6
5
Fig. 1. Effect of Ap, on the decline of v,.
,
0.8
_._
2.39/i :s.-. ._"..__._ .,,__x
q,=*s~lo-~in~'.~ -
--..._ __, _
! i...-. 2.36;
.
,,"___
,,,=L.7x/O-"M3's ^. ..-_ ...I__
J IS
Fig. 3. Effect of q, on the decline of v,,,.
I
H.-M. Yeh / Desalination 145 (2002) 153-157
156
2.36; : q=a.j@*rw ; L.=4ncm
a& =l.u~~lcJ" h $=t.$*l@*,,,"ri
2.36L_ .....A 0.0
! 0.2
Fig. 4. Effect of r, on the
0.4
L_-.-L-_~_._~__~ z
0.6
0.6
1.c
decline of v,.
rapidly for larger Ci because in this case, the thickness of concentration polarization layer increases more seriously along the tube. We may find in Figs. 3 and 4 that for larger qi, or for smaller r,,,,v, is larger at the tube entrance, but then declines more rapidly along the tube. This is because that while v, increases with the increase of fluid velocity by increasing q,, or by decreasing r,, the increase of fluid velocity will also lead to a more rapid decline of the effective transmembrane pressure as well as the permeate flux along the tube. Acknowledgements
The author wishes to express his thanks to the National Science Council of ROC for financial aid with the Grant No. NSC89-2214-E-032-009. References PI
PI
M.C. Poter, Membrane filtration, in P.A. Schweitzer, ed., Handbook of Separation Techniques for Chemical Engineers, McGraw-HilL New York, 1979, Sect. 2.1. M. Cheryan, Ultrafiltration Handbook, Technomic Publishing, Lancaster, PA, 1986, Sect. 8.
[3] W.F. Blatt, A. Dravid, AS. Michales and L. Nelsen, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control technique, in: J.E. Filnn, ed., Membrane Science and Technology, Plenum Press, New York, 1970, p. 47. [4] M.C. Poter, Concentration polarization with membrane ultrafiltration, Ind. Eng. Chem Proc. Res. Dev., 11 (1972) 234. R.B. Grieves, D. Bhattacharyya, W.G.. Schomp and PI J.L. Bewley, Membrane ultrafiltration of a nonionic surfactant, AIChE J., 19 (1973) 766. [61J.J.S. Shen and R.F Probstein, On the prediction of limiting flux in laminar ultrafiltration of macromolecular solution, Ind. Eng. Chem. Fundam., 16 (1977) 459. [71 S. Nokao, T. Nomura and S. Kimura, Characteristics of macromolecular gel layer formed on ultrafiltration tubular membrane, AIChE J., 25 (1979) 615. PI A.G. Fane, C.J.D Fell and A.G. Waters, The relationship between membrane surface pore characteristics and flux for ultrafiltration membranes, J. Membr. Sci., 9 (1981) 245. A.G. Fane, Ultrafiltration of suspensions, J. Membr. r91 Sci., 20 (1984) 249. DOI J.G. Wijmans, S. Nakao and C.A. Smolders, Flux limitation in ultrafiltration: osmotic pressure model and gel layer model, J. Membr. Sci., 20 (1984) 115. Pll A.A. Kozinski and E.N. Lightfoot, Protein ultrafiltration: a general example of boundary layer filtration, AIChE J., 18 (1972) 1030. WI W. Leung and R.F. Probstein, Low polarization in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fundam., 18 (1979) 274. [I31 R.P. Wendt, E. Klein, F.F. Holland and K.E. Eberle, Hollow fiber ultrafiltration of calf serum and albumin in the pregel uniform-wall-flux region, Chem. Eng. Commum., 8 (1981) 251. [I41 S. Nakao and S. Kimura, Models of membrane transport phenomena and their applications for ultrafiltration data, J. Chem. Eng. Jpn., 15 (1982) 200. I.151 C. Kleinstreuer and MS. Paller, Laminar dilute suspension flows in plate-and-frame ultrafiltration units, AIChE J., 29 (1983) 529. [I61 M.J. Clifton, N. Abidine, P. Aptel and V. Sanchez, Growth ofthe polarization layer in ultrafiltration with hollow-fiber membranes, J. Membr. Sci., 21 (1984) 233.
H.-M. Yeh /Desalination [17] R.P. Gooding Ma, C.H. Gooding and W.K. Alex-
ander, A dynamic model for low-pressure, hollowfiber ultrafiltration, AIChE J., 3 1 (1985) 1728. [18] H. Nabetani, M. Nakajima, A. Watanabe, S. Nakao and S. Kumura, Effects of osmotic pressure and adsorption on ultrafiltration of ovalbumin, AIChE J., 36 (1990) 907. [ 191 B.H. Chiang and M. Cheryan, Ultrafiltration on skim milk in hollow fibers, J. Food Sci., 5 1 (1986) 340. [20] M. Assadi and D.A. White, A mode for determining the steady-state flux of inorganic microfiltration membrane, Chem. Eng. J., 48 (1992) 11. [21] H.M. Yeh and T.W. Cheng, Resistance-in-series of membrane ultrafiltration in hollow fiber of tube-andshell arrangement, Sep. Sci. Technol., 28 (1993) 1341. [22] H.M. Yeh and H.H. Wu, Membrane ultrafiltration in combined hollow-fiber module systems, J. Membr. Sci., 124 (1997) 93. [23] H.M. Yeh and J.W. Tsai, Membrane ultrafiltration in multipass hollow-fiber modules, J. Membr. Sci., 142 (1998) 61.
145 (2002) 153-157
157
[24] H.M. Yeh, H.Y. Chen and K.T. Chen, Membrane ultrafIltration in a tubular module with a steel rod inserted concentrically for improved performance, J. Membr. Sci., 168 (2000) 121. [25] H.M. Yeh and K.T. Chen, Improvement of ultrafiltration performance in tubular membranes using a twisted wire-rod assembly, J. Membr. Sci., 178 (2000) 43. [26] H.M. Yeh, J.F. Dong, M.J. Hsieh and C.C. Yang, Prediction ofpermeate flux for ultrafiltration in wirerod tubular-membrane modules, J. Membr. Sci., 5225 (2002). [27] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 197 1, p. 5 1. [28] J.F. Dong, Permeate flux analysis for membrane ultrafiltration, M.S. Thesis, Tamkang University, Tamsui, Taiwan, ROC, 2001, p. 32. [29] T.W. Cheng, A study on the hollow-fiber membrane ultrafiltration, Ph.D. Thesis, National Taiwan University, Taipei, Taiwan, ROC, 1992, p. 146.
ELSEVIER
www.elsevier.com/locate/desal
Decline of permeate flux for ultrafiltration along membrane tubes Ho-Ming Yeh Department of Chemical Engineering, Tamkang University Tamsui, Taipei, Taiwan 2.51, ROC Tel. +886 (02) 2621-5656, ext. 2601; Fax +886 (02) 262-0987; email: [email protected] Received 8 February 2002; accepted 22 February 2002
Abstract The decline of permeate flux for ultrafiltration along membrane tubes was predicted from mass and momentum balances, based on the resistance-in-series model coupled with the considerations of the declinations of flow rate and transmembrane pressure along the membrane tubes. It is concluded that the declination of permeate flux occurs mainly due to the decline of transmembrane pressure and the increase of the thickness of concentration polarization layer along the membrane tube, and thus it is rather sensitive for lower transmembrane pressure and larger solution concentration. Kcyworak
Ultrafiltration; Permeate flux declination; Membrane tube
1. Introduction The ultrafiltration (UF) process was initially developed for the treatment of waste waters and sewage to remove particulate and macromolecular materials. Nowadays, it is applied in a wide variety of fields, from the chemical industry (including electrocoat paint recovery, latex processing, textile size recovery and recovery of lubricating oil) to medical applications (such as kidney dialysis operation), and even to biotechnology (including concentration of milk, egg white, juice, pectin and sugar, and recovery of Presented at the International July 7-12, 2002. 001 l-9164/02/$-
Congress on Membranes
protein from cheese whey, animal blood, gelatin and glue) [ 1,2].
2. Theory A number of mathematical models for permeation analysis of membrane UF are available in the literature that attempt to describe the mechanism oftransport through membranes. The permeate flux of UF of micromolecular solutions is usually analyzed by following models: the gel polarization model [3-91, the osmotic pressure and Membrane
Processes
See front matter 0 2002 Elsevier Science B.V. All rights reserved
PII: SO0 I1 -9 I64(02)0040
I-O
(ICOA4,),Toulouse, France,
154
H.-M. Yeh / Desalination I45 (2002) 153-l 5 7
model [10-l 81,and the resistance-in-series model [19-261. 2.1. Resistance-in-series model In the resistance-in-series model, permeate flux v,(z) may be expressed as
tube, it can be assumed that the momentum balance for laminar flow inside the membrane tube is simply given by the Hagen-Poiseuille equation [27]
_dAp -=dz
vm(z) =
&p(z) R, +Rf+ 4 Ap(z)
(1)
In Eq. (l), R, denotes the intrinsic resistance, and $Ap(z> and Rf are, respectively, the resistance due to the concentration polarization/gel layer with 4 as a proportional constant [22,23] and those due to other fouling phenomena such as solute adsorption, while Ap(x) is the transmembrane pressure defined as
8w 7c
rf
where lo is the viscosity of solution. 2.4. Permeate flux The distribution of permeate flux v,(z) was derived by solving Eqs. (l), (3) and (4) with the following inlet conditions: q=ql
and
Ap=Apiatz=O
(5), (6)
The result is [26] Ap(z) =p(z)-P,
(2)
In the above equation, p(z) is the pressure distribution of the tube side along the axial direction, z, andp, is the permeate pressure of the shell side which may be assumed to be a constant. The resistances R, and R/as well as the proportional constant 4 in Eq. (1) are usually determined experimentally.
Ap
v=
(7)
1 +pAP
in which AP is determined from
2.2. Mass balance Consider a membrane tube of inside radius rm and tube length L. Let q(z) be the volumetric flow rate of solution flowing in the tube side and v,(z) be the permeate flux by UF. Then, a mass balance over a slice dz of the tube gives
dq= -2n; Y,V, dz
2.3. Momentum balance Since the permeation rate of solvent through the membrane wall is small compared with the volumetric flow rate of solution flowing in the
and V and AP are defined as v=
6pm+Rf)v=‘, ‘P A,p=
APi where
Gi
(9), (IO)
H. -M. Yeh / Desalination 145 (2002) 153-1.57
3. Discussion
and conclusions
For calculation of the permeate flux of water from aqueous dextran T500 through a tubular ceramic membrane, let us use the correlation equation of $ and (R, + R/> derived by Dong
P81. 4 (s/m) = 1.37 x 10’ uiWo.02 Cz~‘s3s
(15)
membrance pressure (Api>,solution concentration (C,), volumetric flow rate (q,), and tube radius (r,) as parameters, were calculated from Eq. (7), and some of the results are shown in Figs. 14, respectively. It is seen in Fig. 1 that v,,, declines more rapidly for lower Api due to the serious decline of the effective driving force (transmembrane pressure) in the downstream part of the tube. As shown in Fig. 2, v,,,declines rather 1.5;
(R,,,+ Rf)(Pa-s/m)
155
.
= 1.055~ 10”
.
“__. ._. ___-.-
L
+.__
_l...-_..”_...... -..-_l_ll_~
_
.
c,
=Ecl WI% ..__ _._
.‘
(16) + 1.74 x 10’0 uq~W5Cf.244 where the feed velocity U,is related to the volumetric flow rate of feed solution q, as u, = qilnr,jf. The correlation predictions for the distribution of permeate flux were calculated from Eq. (7) with the use of the empirical equation for the viscosity of dextran T500 aqueous solution [29]. ~1(kg/m’s) = 0.894x 10e3 exp(0.408Ci)
f
“+,
0.5
%rmw%; \
\
/
i /
(17)
‘.. _?(
.‘\\i
=I
‘*
0.0’
The decline of permeate flux v,(z) along the flow channel of tubular membranes with trans-
tcxltPR q,ts2%*w*tn3.+
1,=4Dcm r, =0,3mt 0.2
0.0
0.4
2:
0.6
0.8
l.C
Fig. 2. Effect of Ci on the decline of v,,,. 2.40,~ .__..._...._... .._._._._..__._ _.
2 "E 12 3 ", Y E
i 0.0’
i
0.0
.
0.2
i...
i.
0.4
0.6
5
Fig. 1. Effect of Ap, on the decline of v,.
,
0.8
_._
2.39/i :s.-. ._"..__._ .,,__x
q,=*s~lo-~in~'.~ -
--..._ __, _
! i...-. 2.36;
.
,,"___
,,,=L.7x/O-"M3's ^. ..-_ ...I__
J IS
Fig. 3. Effect of q, on the decline of v,,,.
I
H.-M. Yeh / Desalination 145 (2002) 153-157
156
2.36; : q=a.j@*rw ; L.=4ncm
a& =l.u~~lcJ" h $=t.$*l@*,,,"ri
2.36L_ .....A 0.0
! 0.2
Fig. 4. Effect of r, on the
0.4
L_-.-L-_~_._~__~ z
0.6
0.6
1.c
decline of v,.
rapidly for larger Ci because in this case, the thickness of concentration polarization layer increases more seriously along the tube. We may find in Figs. 3 and 4 that for larger qi, or for smaller r,,,,v, is larger at the tube entrance, but then declines more rapidly along the tube. This is because that while v, increases with the increase of fluid velocity by increasing q,, or by decreasing r,, the increase of fluid velocity will also lead to a more rapid decline of the effective transmembrane pressure as well as the permeate flux along the tube. Acknowledgements
The author wishes to express his thanks to the National Science Council of ROC for financial aid with the Grant No. NSC89-2214-E-032-009. References PI
PI
M.C. Poter, Membrane filtration, in P.A. Schweitzer, ed., Handbook of Separation Techniques for Chemical Engineers, McGraw-HilL New York, 1979, Sect. 2.1. M. Cheryan, Ultrafiltration Handbook, Technomic Publishing, Lancaster, PA, 1986, Sect. 8.
[3] W.F. Blatt, A. Dravid, AS. Michales and L. Nelsen, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control technique, in: J.E. Filnn, ed., Membrane Science and Technology, Plenum Press, New York, 1970, p. 47. [4] M.C. Poter, Concentration polarization with membrane ultrafiltration, Ind. Eng. Chem Proc. Res. Dev., 11 (1972) 234. R.B. Grieves, D. Bhattacharyya, W.G.. Schomp and PI J.L. Bewley, Membrane ultrafiltration of a nonionic surfactant, AIChE J., 19 (1973) 766. [61J.J.S. Shen and R.F Probstein, On the prediction of limiting flux in laminar ultrafiltration of macromolecular solution, Ind. Eng. Chem. Fundam., 16 (1977) 459. [71 S. Nokao, T. Nomura and S. Kimura, Characteristics of macromolecular gel layer formed on ultrafiltration tubular membrane, AIChE J., 25 (1979) 615. PI A.G. Fane, C.J.D Fell and A.G. Waters, The relationship between membrane surface pore characteristics and flux for ultrafiltration membranes, J. Membr. Sci., 9 (1981) 245. A.G. Fane, Ultrafiltration of suspensions, J. Membr. r91 Sci., 20 (1984) 249. DOI J.G. Wijmans, S. Nakao and C.A. Smolders, Flux limitation in ultrafiltration: osmotic pressure model and gel layer model, J. Membr. Sci., 20 (1984) 115. Pll A.A. Kozinski and E.N. Lightfoot, Protein ultrafiltration: a general example of boundary layer filtration, AIChE J., 18 (1972) 1030. WI W. Leung and R.F. Probstein, Low polarization in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fundam., 18 (1979) 274. [I31 R.P. Wendt, E. Klein, F.F. Holland and K.E. Eberle, Hollow fiber ultrafiltration of calf serum and albumin in the pregel uniform-wall-flux region, Chem. Eng. Commum., 8 (1981) 251. [I41 S. Nakao and S. Kimura, Models of membrane transport phenomena and their applications for ultrafiltration data, J. Chem. Eng. Jpn., 15 (1982) 200. I.151 C. Kleinstreuer and MS. Paller, Laminar dilute suspension flows in plate-and-frame ultrafiltration units, AIChE J., 29 (1983) 529. [I61 M.J. Clifton, N. Abidine, P. Aptel and V. Sanchez, Growth ofthe polarization layer in ultrafiltration with hollow-fiber membranes, J. Membr. Sci., 21 (1984) 233.
H.-M. Yeh /Desalination [17] R.P. Gooding Ma, C.H. Gooding and W.K. Alex-
ander, A dynamic model for low-pressure, hollowfiber ultrafiltration, AIChE J., 3 1 (1985) 1728. [18] H. Nabetani, M. Nakajima, A. Watanabe, S. Nakao and S. Kumura, Effects of osmotic pressure and adsorption on ultrafiltration of ovalbumin, AIChE J., 36 (1990) 907. [ 191 B.H. Chiang and M. Cheryan, Ultrafiltration on skim milk in hollow fibers, J. Food Sci., 5 1 (1986) 340. [20] M. Assadi and D.A. White, A mode for determining the steady-state flux of inorganic microfiltration membrane, Chem. Eng. J., 48 (1992) 11. [21] H.M. Yeh and T.W. Cheng, Resistance-in-series of membrane ultrafiltration in hollow fiber of tube-andshell arrangement, Sep. Sci. Technol., 28 (1993) 1341. [22] H.M. Yeh and H.H. Wu, Membrane ultrafiltration in combined hollow-fiber module systems, J. Membr. Sci., 124 (1997) 93. [23] H.M. Yeh and J.W. Tsai, Membrane ultrafiltration in multipass hollow-fiber modules, J. Membr. Sci., 142 (1998) 61.
145 (2002) 153-157
157
[24] H.M. Yeh, H.Y. Chen and K.T. Chen, Membrane ultrafIltration in a tubular module with a steel rod inserted concentrically for improved performance, J. Membr. Sci., 168 (2000) 121. [25] H.M. Yeh and K.T. Chen, Improvement of ultrafiltration performance in tubular membranes using a twisted wire-rod assembly, J. Membr. Sci., 178 (2000) 43. [26] H.M. Yeh, J.F. Dong, M.J. Hsieh and C.C. Yang, Prediction ofpermeate flux for ultrafiltration in wirerod tubular-membrane modules, J. Membr. Sci., 5225 (2002). [27] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, 197 1, p. 5 1. [28] J.F. Dong, Permeate flux analysis for membrane ultrafiltration, M.S. Thesis, Tamkang University, Tamsui, Taiwan, ROC, 2001, p. 32. [29] T.W. Cheng, A study on the hollow-fiber membrane ultrafiltration, Ph.D. Thesis, National Taiwan University, Taipei, Taiwan, ROC, 1992, p. 146.